Time Complexity of building a heap

Consider the following algorithm for building a Heap of an input array A.

BUILD-HEAP(A)
heapsize := size(A);
for i := floor(heapsize/2) downto 1
do HEAPIFY(A, i);
end for
END

A quick look over the above algorithm suggests that the running time is , since each call to Heapify costs and Build-Heap makes such calls.
This upper bound, though correct, is not asymptotically tight.

We can derive a tighter bound by observing that the running time of Heapify depends on the height of the tree ‘h’ (which is equal to lg(n), where n is number of nodes) and the heights of most sub-trees are small.
The height ’h’ increases as we move upwards along the tree. Line-3 of Build-Heap runs a loop from the index of the last internal node (heapsize/2) with height=1, to the index of root(1) with height = lg(n). Hence, Heapify takes different time for each node, which is .

For finding the Time Complexity of building a heap, we must know the number of nodes having height h.
For this we use the fact that, A heap of size n has at most nodes with height h.

Now to derive the time complexity, we express the total cost of Build-Heap as-

(1)

Step 2 uses the properties of the Big-Oh notation to ignore the ceiling function and the constant 2(). Similarly in Step three, the upper limit of the summation can be increased to infinity since we are using Big-Oh notation.

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